Cárdenas M. L. C., Letelier J.-C., Gutierrez C., Cornish-Bowden A. & Soto-Andrade J. (2010) Closure to efficient causation, computability and artificial life. Journal of Theoretical Biology 263(1): 79–92. https://cepa.info/3631

The major insight in Robert Rosen’s view of a living organism as an (M, R)-system was the realization that an organism must be “closed to efficient causation”, which means that the catalysts needed for its operation must be generated internally. This aspect is not controversial, but there has been confusion and misunderstanding about the logic Rosen used to achieve this closure. In addition, his corollary that an organism is not a mechanism and cannot have simulable models has led to much argument, most of it mathematical in nature and difficult to appreciate. Here we examine some of the mathematical arguments and clarify the conditions for closure.

Díaz-Rojas D. & Soto-Andrade J. (2017) Enactive metaphors in mathematical problem solving. In: Doole T. & Gueudet G. (eds.) Proceedings of CERME10. Dublin, Ireland: 3904–3911. https://cepa.info/6173

We are interested in exploring the role of enactive metaphoring in mathematical thinking, especially in the context of problem posing and solving, not only as a means to foster and enhance the learner’s ability to think mathematically but also as a mean to alleviate the cognitive abuse that the teaching of mathematics has turned out to be for most children and adolescents in the world. We present some illustrative examples to this end besides describing our theoretical framework.

This article analyses the work of Robert Rosen on an interpretation of metabolic networks that he called (M, R) systems. His main contribution was an attempt to prove that metabolic closure (or metabolic circularity) could be explained in purely formal terms, but his work remains very obscure and we try to clarify his line of thought. In particular, we clarify the algebraic formulation of (M, R) systems in terms of mappings and sets of mappings, which is grounded in the metaphor of metabolism as a mathematical mapping. We define Rosen’s central result as the mathematical expression in which metabolism appears as a mapping f that is the solution to a fixed-point functional equation. Crucially, our analysis reveals the nature of the mapping, and shows that to have a solution the set of admissible functions representing a metabolism must be drastically smaller than Rosen’s own analysis suggested that it needed to be. For the first time, we provide a mathematical example of an (M, R) system with organizational invariance, and we analyse a minimal (three-step) autocatalytic set in the context of (M, R) systems. In addition, by extending Rosen’s construction, we show how one might generate self-referential objects f with the remarkable property f(f)=f, where f acts in turn as function, argument and result. We conclude that Rosen’s insight, although not yet in an easily workable form, represents a valuable tool for understanding metabolic networks.

Soto-Andrade J. (2018) Enactive metaphorising in the learning of mathematics. In: Kaiser G., Forgasz H., Graven M., Kuzniak A., Simm E. & Xu B. (eds.) Invited lectures from the 13th International Congress on Mathematical Education. Springer, Cham: 619–638. https://cepa.info/6219

We argue that an approach to the learning of mathematics based on enactive (bodily acted out) metaphorising may significantly help in alleviating the cognitive abuse millions of children worldwide suffer when exposed to mathematics. We present illustrative examples of enactive metaphoric approaches in the context of problem posing and solving in mathematics education, involving geometry and randomness, two critical subjects in school mathematics. Our examples show to what extent the way a mathematical situation is metaphorised and enacted by the learners shapes their emerging ideas and insights and how this may help to bridge the gap between the ‘mathematically gifted’ and those apparently not so gifted or mathematically inclined. Our experimental background includes a broad spectrum of prospective secondary math teachers, in-service primary teachers and their pupils, first-year university students majoring in social sciences and humanities and university students majoring in mathematics.

We consider an extension of Lawvere’s Theorem showing that all classical results on limitations (i.e. Cantor, Russel, Gödel, etc.) stem from the same underlying connection between self-referentiality and fixed points. We first prove an even stronger version of this result. Secondly, we investigate the Theorem’s converse, and we are led to the conjecture that any structure with the fixed point property is a retract of a higher reflexive domain, from which this property is inherited. This is proved here for the category of chain complete posets with continuous morphisms. The relevance of these results for computer science and biology is briefly considered.

Soto-Andrade J. & Varela F. J. (1990) On mental rotations and cortical activity patterns: A linear representation is still wanted. Biological Cybernetics 64: 221–223.

Soto-Andrade J. & Yañez-Aburto A. (2019) Acknowledging the Ouroboros: An enactivist and metaphoric approach to problem solving. In: Felmer P., Koichu B. & Liljedahl P. (eds.) Problem solving in mathematics instruction and teacher professional development. Springer, Berlin: 67–85. https://cepa.info/6708

We are interested in exploring and developing an enactivist approach to problem posing and problem solving. We use here the term “enactivist approach” to refer to Varela’s radically nonrepresentationalist and pioneering “enactive approach to cognition” (Varela et al., The embodied mind: Cognitive science and human experience. Cambridge, MA: The MIT Press, 1991), to avoid confusion with the enactive mode of representation of Bruner, which is still compatible with a representationalist view of cognition. In this approach, problems are not standing “out there” waiting to be solved, by a solver equipped with a suitable toolbox of strategies. They are instead co-constructed through the interaction of a cognitive agent and a milieu, in a circular process well described by the metaphor of the Ouroboros (the snake eating its own tail). Also, cognition as enaction is metaphorized by Varela as “lying down a path in walking.” In this vein, we present here some paradigmatic examples of enactivist, and metaphorical, approaches to problem solving and problem posing, involving geometry, algebra, and probability, drawn from our didactical experimenting with a broad spectrum of learners, which includes humanities-inclined university students as well as prospective and in-service maths teachers. Our examples may be metaphorized as cognitive random walks in the classroom, stemming and unfolding from a situational seed.

Soto-Andrade J. & Yáñez-Aburto A. (2019) Problematizing as an Avatar of Mathematical Activity: Replications and Prospects. Constructivist Foundations 15(1): 71–73. https://cepa.info/6167

Open peer commentary on the article “Problematizing: The Lived Journey of a Group of Students Doing Mathematics” by Robyn Gandell & Jean-François Maheux. Abstract: We discuss mathematical problematizing as presented in the target article, the perspective of which we mostly share, and compare it with the case of in-service primary-school teachers engaged in the same mathematical activity. Also, we comment on related previous experimental research conducted in our research group, which suggests further developments in the spirit of the target article.